Algorithms & Data Structures
the bound is a claim, and it is checkable
Concept

Pigeonhole principle — where it appears

If more things are placed than there are places, some place holds two. It is the whole of several lower bounds here: a summary with fewer states than answers must give two different inputs the same answer, and a transform with fewer outputs than inputs must lose something.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

H₀ = 3.8916642561k4k16kwindow, in symbolsbits per symbol0.04.38.6LZSSmean symbols covered per match4.65.15.76.98.79.6model: a window of recent text, no probabilities2.13 bits/symbol at a window of 16,384

The dictionary that builds itself

LZSS contains no probability, no frequency table and no entropy calculation. Its entire model is a window of recent text and its only move is to say "the next nine symbols are the ones that appeared 1,200 positions ago". On a stream whose zeroth-order floor is 3.89 bits per symbol it spends 2.11, and widening its window past 4,096 makes it worse rather than better.

practice · Bits thousands of bits · the floor is 31.9kthe text as it is34.0k streamto undo itnothinga shuffle agreed in advance34.4k streamto undo itnothingthe characters sorted0.3k streamto undo it31.8kthe Burrows–Wheeler transform14.8k streamto undo itnothingmodel: order 0 after move-to-front · Words from a fixed vocabularydashed: coding the text as it stands

What a reordering costs to undo

Sorting the characters of a text clusters them perfectly: a move-to-front pass then leaves 289 bits where the text's own floor is 31,931. Naming which arrangement of those characters the text was costs 31,827 bits, and the two numbers add to the floor it started from. The Burrows–Wheeler transform clusters less and costs nothing to undo, which is the only reason it is the one that is used.

floors · Bits

Named alongside it

The objects these essays reach for when they reach for this one.

Bits per symbolBurrows-wheeler transformCompression ratioCounting argumentDeflateDictionary codingEntropyInformation-theoretic boundInvertibilityLower boundLz77Lz78

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